The Haldane phase is a celebrated example of a magnetic topological phase but rarely reported in materials. The authors show here an NMR study on the spin-1 chain magnet NiC
2O
4\ifmmode\cdot\else\textperiodcentered\fi{}2NH
3, which determines its ground state to be a Haldane phase at zero field. An applied magnetic field induces a magnetic ordering at low temperatures and a Tomonaga-Luttinger liquid phase above the ordering temperature with attractive fermionic interactions.
We probe the magnetic field-induced Tomonaga-Luttinger liquid (TLL) state in the bond-alternating spin-
1/2 antiferromagnetic (AFM) chain compound
NaVOPO4 using thermodynamic as well as local $\ensuremath{\mu}\mathrm{SR}$ and
31P NMR probes down to mK temperatures in magnetic fields up to 14 T. The $\ensuremath{\mu}\mathrm{SR}$ and NMR relaxation rates in the gapless TLL regime decay slowly following characteristic power-law behavior, enabling us to directly determine the interaction parameter
K as a function of the magnetic field. These estimates are crosschecked using magnetization and specific heat data. The field-dependent
K lies in the range of
0.4<K<1 and indicates the repulsive nature of interactions between the spinless fermions, in line with the theoretical predictions. This renders
NaVOPO4 the first experimental realization of TLL with repulsive fermionic interactions in hitherto studied
S=1/2 bond-alternating AFM-AFM chain compounds.
A bosonized-field-theory representation of spin operators in the uniform-field-induced Tomonaga–Luttinger-liquid (TLL) phase in spin-1 Haldane chains is formulated by means of the non-Abelian (Tsvelik's Majorana fermion theory) and the Abelian bosonizations and Furusaki–Zhang technique. It contains massive magnon fields as well as massless boson fields. From it, asymptotic forms of spin correlation functions in the TLL phase are completely determined. Applying the formula, we further discuss the effects of perturbations (bond alternation, single-ion anisotropy terms, staggered fields, etc) for the TLL state, and the string order parameter. Throughout the paper, we often consider in some detail how the symmetry operations of the Haldane chains are translated in the low-energy effective theory.